Algebraic optimisation of excavation alignment using the stress tensor

A simple algebraic method of aligning large scale underground excavations such as a crusher station or workshop or indeed any large cavern or mine layout to the optimal stress condition is outlined. The method uses the stress transformation law to determine how the component stresses vary as the stress tensor is rotated around its axis. The aim is to define the orientations at which the maximum and minimum horizontal stresses occur such that the excavation can be aligned with its long axis paralleling the trend of the maximum horizontal stress leading to the minimum potential for overstressing of the side walls. The physical environment in which the stress field is acting is unchanged. At the design location, the trend and plunge of the major principal stress is unchanged. However, the coordinate system by which the location of the stress field is described is arbitrary and the stress transformation law allows that coordinate system to be rotated to a more convenient orientation and for the component stresses to be calculated within the alterative coordinate system. The stress field is typically described in terms of normal and shear stresses aligned to a north–south, east–west, vertical, coordinate system. In this scenario, there are two orientations in the 360° horizontal plane defined by the coordinate system where the horizontal normal stresses are equal, and at 45° to those orientations, two orientations where they diverge to the maximum extent. Those orientations of maximum divergence correspond to the trend of the maximum and minimum horizontal stresses, usually designated σH and σh, and that is what the algebraic manipulation seeks to determine. This technique is a first pass assessment which relies on a knowledge of the full stress tensor. It is not a replacement for numerical modelling or detailed design, but a tool to guide planning.